Question

to solve an equation.
$\frac{e}{4} = 2.5$
$\frac{e}{4} cdot 4 = 2.5 cdot 4$
setting up
$e = 8$
calculating
where did gary make his first mistake?
choose 1 answer.

Explanation:

Step1: Recall the equation-solving rule

To solve for \( e \) in \( \frac{e}{4}=2.5 \), we should multiply both sides by 4. The correct step for setting up the solution is \( \frac{e}{4}\times4 = 2.5\times4 \), which simplifies to \( e=10 \).

Step2: Analyze Gary's mistake

Gary's setting up step is \( \frac{e}{4}\cdot4 = 2.5\cdot4 \)? Wait, no, looking at the given, Gary's setting up step (the middle one) is \( \frac{e}{4}\cdot4 = 2.5\cdot4 \)? Wait, the original equation is \( \frac{e}{4}=2.5 \). To isolate \( e \), we multiply both sides by 4. So left side: \( \frac{e}{4}\times4 = e \), right side: \( 2.5\times4 = 10 \). But Gary's setting up step (the middle expression) is \( \frac{e}{4}\cdot4 = 2.5\cdot4 \)? Wait, no, the middle line shows \( \frac{e}{4}\cdot4 = 2.5\cdot4 \)? Wait, the user's image (after OCR) has: first line \( \frac{e}{4}=2.5 \), second line \( \frac{e}{4}\cdot4 = 2.5\cdot4 \)? Wait, no, the second line in the image (from OCR) is \( \frac{e}{4}\cdot4 = 2.5\cdot4 \)? Wait, no, the OCR shows "e/4 .4 = 2.5.4" (probably a typo, maybe \( \frac{e}{4}\cdot4 = 2.5\cdot4 \) is miswritten, but maybe Gary's mistake is in the setting up: when solving \( \frac{e}{4}=2.5 \), to eliminate the denominator 4, we multiply both sides by 4. So the correct operation is \( \frac{e}{4}\times4=2.5\times4 \). But if Gary wrote \( \frac{e}{4}\cdot4 = 2.5\cdot4 \) as \( \frac{e}{4}\cdot4 = 2.5\cdot4 \), but maybe the mistake is in the setting up step: he might have multiplied the right side by 4 incorrectly or there's a typo. Wait, the calculating step: he got \( e = 8 \), but \( 2.5\times4 = 10 \), so the mistake is in the calculating? Wait, no, the question is "Where did Gary make his first mistake?". Let's re - examine:

Original equation: \( \frac{e}{4}=2.5 \)

To solve for \( e \), multiply both sides by 4:

Left side: \( \frac{e}{4}\times4=e \)

Right side: \( 2.5\times4 = 10 \)

Gary's setting up step (the middle line) is \( \frac{e}{4}\cdot4 = 2.5\cdot4 \)? Wait, maybe the middle line is a mis - write, and actually, Gary's mistake is in the setting up: if he thought to multiply the right side by 4 but miscalculated the operation, or maybe in the setting up, he should have multiplied both sides by 4, but if the middle line is \( \frac{e}{4}\cdot4 = 2.5\cdot4 \), but then in calculating, he got \( e = 8 \) instead of \( 10 \). Wait, no, the first mistake: the setting up step. Wait, the original equation is \( \frac{e}{4}=2.5 \). The correct way to eliminate the denominator is to multiply both sides by 4. So the setting up step should be \( \frac{e}{4}\times4=2.5\times4 \). If Gary's setting up step (the middle line) is written as \( \frac{e}{4}\cdot4 = 2.5\cdot4 \), but maybe he made a mistake in the operation on the right side. Wait, \( 2.5\times4 = 10 \), but if he thought to do \( 2.5 + 4 \) or something else, but no, the first mistake is in the setting up? Wait, no, let's check the calculating step. Wait, \( 2.5\times4 = 10 \), but Gary got \( e = 8 \), so he made a mistake in calculating \( 2.5\times4 \). But the question is about the first mistake. The first step after the original equation is the setting up (multiplying both sides by 4) and then calculating. Wait, the original equation is \( \frac{e}{4}=2.5 \). The setting up step: multiply both sides by 4. So \( \frac{e}{4}\times4=2.5\times4 \). If Gary's setting up step is \( \frac{e}{4}\cdot4 = 2.5\cdot4 \), that's correct for the operation, but then in calculating, he did \( 2.5\times4 = 8 \) which is wrong. Wait, no, \( 2.5\times4=10 \). So the first mistake is in the calculating? Wait, no, the setting up step: maybe the way he wrote the equation. Wait, the user's image shows:

First line: \( \frac{e}{4}=2.5 \)

Second line: \( \frac{e}{4}\cdot4 = 2.5\cdot4 \) (maybe a typo, but let's assume)

Third line: \( e = 8 \)

So the mistake is in the calculating step? Wait, no, \( 2.5\times4 = 10 \), not 8. But the question is "Where did Gary make his first mistake?". The first step after the original equation is the setting up (multiplying both sides by 4) and then calculating. Wait, maybe the setting up step: he should have multiplied both sides by 4, but if the middle line is \( \frac{e}{4}\cdot4 = 2.5\cdot4 \), that's correct, but then the calculating is wrong. But maybe the setting up step is wrong. Wait, no, the correct operation to solve \( \frac{e}{4}=2.5 \) is to multiply both sides by 4. So the setting up step is correct in operation, but the calculating is wrong. But the options are "Setting up" or "Calculating". Wait, the problem says "Choose 1 answer" between setting up or calculating.

Wait, let's re - do the solution:

Equation: \( \frac{e}{4}=2.5 \)

To solve for \( e \), multiply both sides by 4:

\( \frac{e}{4}\times4=2.5\times4 \)

Simplify left side: \( e \)

Simplify right side: \( 2.5\times4 = 10 \)

So \( e = 10 \)

Gary's steps:

Setting up: \( \frac{e}{4}\cdot4 = 2.5\cdot4 \) (this is correct in terms of the operation, multiply both sides by 4)

Calculating: \( e = 8 \) (this is wrong, because \( 2.5\times4 = 10 \), not 8)

Wait, but maybe the setting up step is wrong. Wait, maybe the original equation is \( \frac{e}{4}=2.5 \), and Gary wrote \( \frac{e}{4}\cdot4 = 2.5\cdot4 \) as \( \frac{e}{4}\cdot4 = 2.5\cdot4 \), but maybe he misapplied the multiplication. Wait, no, the first mistake is in the calculating step, because the setting up step (multiplying both sides by 4) is the correct operation, but the calculation of \( 2.5\times4 \) is wrong. But the problem says "first mistake". Wait, maybe the setting up step: he should have multiplied both sides by 4, but if the middle line is \( \frac{e}{4}\cdot4 = 2.5\cdot4 \), that's correct. Wait, maybe the user made a typo, and the middle line is \( \frac{e}{4}+4 = 2.5 + 4 \), but no, the OCR shows \( \frac{e}{4}\cdot4 = 2.5\cdot4 \). Alternatively, maybe the first mistake is in the setting up: he should have multiplied both sides by 4, but he did \( \frac{e}{4}\cdot4 = 2.5\cdot4 \), but \( 2.5\cdot4 \) is wrong? No, \( 2.5\times4 = 10 \). Wait, maybe the question is that Gary's first mistake is in the calculating step, because he calculated \( 2.5\times4 \) as 8 instead of 10. Or maybe in the setting up, he should have multiplied both sides by 4, but he did something else. Wait, the problem is asking where the first mistake is, between setting up and calculating. Since the setting up step (multiplying both sides by 4) is the correct algebraic operation, the mistake is in the calculating step, where he computed \( 2.5\times4 \) as 8 instead of 10. But wait, maybe the setting up step is wrong. Wait, no, the correct way to solve \( \frac{e}{4}=2.5 \) is to multiply both sides by 4, so the setting up step is correct. The mistake is in the calculating. But the problem says "first mistake". So the first mistake is in the calculating step? Wait, no, the first step after the original equation is the setting up, then calculating. So if the setting up step is correct, the first mistake is in calculating. But maybe the setting up step is wrong. Wait, maybe the original equation is \( \frac{e}{4}=2.5 \), and Gary wrote \( \frac{e}{4}\cdot4 = 2.5\cdot4 \), but \( 2.5\cdot4 \) is not the correct operation. Wait, no, it is the correct operation. I think the mistake is in the calculating step, because \( 2.5\times4 = 10 \), not 8. But the problem says "Choose 1 answer" between setting up or calculating. So the answer is "Calculating" or "Setting up". Wait, maybe the setting up step is wrong. Wait, let's check again. The original equation is \( \frac{e}{4}=2.5 \). To solve for \( e \), we need to isolate \( e \). The coefficient of \( e \) is \( \frac{1}{4} \), so we multiply both sides by the reciprocal, which is 4. So \( e=2.5\times4 \). Gary's setting up step: \( \frac{e}{4}\cdot4 = 2.5\cdot4 \), which is correct. Then his calculating step: \( e = 8 \), which is wrong, because \( 2.5\times4 = 10 \). So the first mistake is in the calculating step. But maybe the problem considers that the setting up step is wrong. Wait, maybe the user made a typo, and the middle line is \( \frac{e}{4}+4 = 2.5 + 4 \), which would be a setting up mistake (adding 4 instead of multiplying by 4). But based on the OCR, the middle line is \( \frac{e}{4}\cdot4 = 2.5\cdot4 \), so the setting up step is correct, and the calculating step is wrong. So the first mistake is in the calculating step. But the problem says "Where did Gary make his first mistake? Choose 1 answer" (between setting up or calculating). So the answer is "Calculating" or "Setting up". Wait, maybe I made a mistake. Let's re - calculate \( 2.5\times4 \): \( 2.5\times4=(2 + 0.5)\times4=2\times4+0.5\times4 = 8 + 2=10 \). Oh! Wait, \( 2\times4 = 8 \) and \( 0.5\times4 = 2 \), so \( 8 + 2 = 10 \). Wait, Gary got \( e = 8 \), which is the result of \( 2\times4 \), ignoring the 0.5. So maybe the mistake is in the setting up? No, the setting up is multiplying both sides by 4. Wait, no, the calculating step: he calculated \( 2.5\times4 \) as 8, which is wrong. So the first mistake is in the calculating step.

Answer:

Calculating (because when solving \( \frac{e}{4}=2.5 \), after setting up \( \frac{e}{4}\cdot4 = 2.5\cdot4 \), the correct calculation of \( 2.5\times4 \) is 10, but Gary got \( e = 8 \), showing a mistake in the calculating step)